Abstract
The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called “noncentral limit theorems”. It shares the same covariance as fractional Brownian motion. We study the asymptotic distribution of the quadratic variations of the Rosenblatt process based on long filters, including filters based on high-order finite-dierence and waveletbased schemes. We find exact formulas for the limiting distributions, which we then use to devise strongly consistent estimators of the self-similarity parameter H. Unlike the case of fractional Brownian motion, no matter now high the filter orders are, the estimators are never asymptotically normal, converging instead in the mean square to the observed value of the Rosenblatt process at time 1.
Highlights
Self-similar stochastic processes are of practical interest in various applications, including econometrics, internet traffic, and hydrology
The fractional Brownian motion is the usual candidate to model phenomena in which the selfsimilarity property can be observed from the empirical data
In addition to being self-similar with parameter H, which is evident from the covariance function, fractional Brownian motion (fBm) has correlated increments: from (1) we get, as n → ∞, E BH (n) − BH (1) BH (1) = H (2H − 1) n2H−2 + o n2H−2 ; (2)
Summary
Self-similar stochastic processes are of practical interest in various applications, including econometrics, internet traffic, and hydrology. The results in these articles, and in the previous references on the non-central limit theorem, have one point in common: of all the Hermite processes, the most important one in terms of limit theorem, apart from fBm, is the Rosenblatt process As such, it should be the first non-Gaussian self-similar process for which to develop a full statistical estimation theory. The normalized T2 always converges (in L2 (Ω)) to a Rosenblatt random variable; the piece that sometimes has normal asymptotics is T4, but since T2 always dominates it, VN ’s behavior is always that of T2 This sort of phenomenon was already noted in [6] with the order-one filter for all non-Gaussian Hermite processes, but we know it occurs for the simplest Hermite process that is not fBm, for filters of all orders.
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